Figure 1

$SU\left(2\right)$ gauge potentials [i.e., charge-density-wave (3CDW) patterns] and their defects. Panels (a) and (b) show the patterns corresponding to ${\zeta }_{x,1}$ and ${\zeta }_{y,2}$ respectively (see Table ); the remaining patterns are related to these by translation. Black (white) circles denote excess positive (negative) charge; larger circles correspond to greater excess charge. Panel (c) sketches a dislocation in the 3CDW pattern; the dislocation core is marked in gray. Dashed red lines trace crests of the 3CDW pattern.Reuse & Permissions

Figure 2

(a) Band structure in the presence of strain, calculated for a 100-site-thick graphene nanoribbon with zigzag edges, showing PLLs. All energies are computed in terms of the unperturbed hopping; in the presence of strain, the hopping along one of the three directions increases linearly from $t$ to $1.9t$ across the sample. The zero-energy level is in fact fourfold degenerate, including two zero-energy PLLs and two modes localized on zigzag edges. (b) Location of Dirac cones in the absence of strain, both with the perturbation ${\zeta }_{x,i}$ (thin solid line) and without it (thick line). Constant gauge potentials (see main text) are expected to shift the Dirac cones. (c) Band structure in the presence of both strain and ${\zeta }_{x,i}$. The fourfold-degenerate zero-energy mode is split; the two more widely separated levels correspond to the lowest PLL. (This was checked by computing the wave functions.)Reuse & Permissions

Figure 3

Pseudo-Landau levels (PLLs) generated by the $SU\left(2\right)$ pseudogauge potential $A_x^2{Q}_2=B{Q}_{2}y$ on a 100-site nanoribbon with zigzag edges. Potential strength varies linearly from zero at one end of the ribbon to $0.74t$ at the other end. Main panel shows wave functions in the lowest three distinct PLLs, calculated on a 100-site nanoribbon with zigzag edges. Inset shows the PLL structure [cf. Fig. 2a], indicating that the PLLs are indeed flat for momenta near the Dirac points.Reuse & Permissions